Some generalizations can be carried out by replacing the Laplace operator in (REF ) by some analogues,
for instance,
the nonlocal diffusion [1]} and the fractional Laplace operators [2]}
which also satisfy the semigroup property
with their discretizations satisfying the analogues of Lemma REF . Furthermore, the proposed stabilizing approaches in this paper
can be naturally extended to many other type of gradient flow problems, which can be handled by the existing SAV schemes.
For example, the fourth-order Cahn–Hilliard equation is the \(H^{-1}\)  gradient flow of the energy functional (REF ),
and satisfies the same energy dissipation law as the Allen–Cahn equation.
The MBP is not valid anymore, but the solution is still \(L^\infty \)  stable.
In the similar spirit of this paper, it is interesting to develop the sESAV schemes for the Cahn–Hilliard equation,
and the discrete \(L^\infty \)  stability of the sESAV solution
can be established by combining the high-order consistency analysis and stability estimate,
as done in [3]}, [4]}, which will also be one of our future works.
